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Mining circuit Lower Bound Proofs for Meta-algorithms

COMPUTATIONAL COMPLEXITY 2015年第2期24卷 333-392页

作者： Chen, Ruiwen Kabanets, Valentine Kolokolova, Antonina Shaltiel, Ronen Zuckerman, David Simon Fraser Univ Sch Comp Sci Burnaby BC V5A 1S6 Canada Mem Univ Newfoundland Dept Comp Sci St John NF A1C 5S7 Canada Univ Haifa Dept Comp Sci IL-31999 Haifa Israel Univ Texas Austin Dept Comp Sci Austin TX 78712 USA

We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for "easy" Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an n-variate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2 (n) ) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size . We get non-trivial compression for functions computable by AC (0) circuits, (de Morgan) formulas, and (read-once) branching programs of the size for which the lower bounds for the corresponding circuit class are known. These compression algorithms rely on the structural characterizations of "easy" functions, which are useful both for proving circuit lower bounds and for designing "meta-algorithms" (such as circuit-sat). For (de Morgan) formulas, such structural characterization is provided by the "shrinkage under random restrictions" results by Subbotovskaya (Doklady Akademii Nauk SSSR 136(3):553-555, 1961) and HAyenstad (SIAM J Comput 27:48-64, 1998), strengthened to the "high-probability" version by Santhanam (Proceedings of the Fifty-First Annual IEEE Symposium on Foundations of Computer Science, pp 183-192, 2010), Impagliazzo, Meka & Zuckerman (Proceedings of the Fifty-Third Annual IEEE Symposium on Foundations of Computer Science, pp 111-119, 2012b), and Komargodski & Raz (Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp 171-180, 2013). We give a new, simple proof of the "high-probability" version of the shrinkage result for (de Morgan) formulas, with improved parameters. We use this shrinkage result to get both compression and #sat algorithms for (de Morgan) formulas of size about n (2). We also use this shrinkage result to get an alternative proof of the result by Komargodski & Raz (Proceedings of the Forty-Fi

关键词： Average-case circuit lower bounds circuit-sat algorithms compression meta-algorithms natural property random restrictions shrinkage of de Morgan formulas

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Mining circuit Lower Bound Proofs for Meta-algorithms

Mining Circuit Lower Bound Proofs for Meta-Algorithms

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IEEE 29th Conference on Computational Complexity (CCC)

作者： Chen, Ruiwen Kabanets, Valentine Kolokolova, Antonina Shaltiel, Ronen Zuckerman, David Simon Fraser Univ Sch Comp Sci Burnaby BC V5A 1S6 Canada Mem Univ Newfoundland Dept Comp Sci St John NF Canada Univ Haifa Dept Comp Sci Haifa Israel Univ Texas Austin Dept Comp Sci Austin TX 78712 USA

ISBN: (纸本)9781479936267

We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for "easy" Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an n-variate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2(n)) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size 2(n)/n. We get non-trivial compression for functions computable by AC(0) circuits, (de Morgan) formulas, and (read-once) branching programs of the size for which the lower bounds for the corresponding circuit class are known. These compression algorithms rely on the structural characterizations of "easy" functions, which are useful both for proving circuit lower bounds and for designing "meta-algorithms" (such as circuit-sat). For (de Morgan) formulas, such structural characterization is provided by the "shrinkage under random restrictions" results [52], [21], strengthened to the "high-probability" version by [48], [26], [33]. We give a new, simple proof of the "high-probability" version of the shrinkage result for (de Morgan) formulas, with improved parameters. We use this shrinkage result to get both compression and #sat algorithms for (de Morgan) formulas of size about n(2). We also use this shrinkage result to get an alternative proof of the recent result by Komargodski and Raz [33] of the average-case lower bound against small (de Morgan) formulas. Finally, we show that the existence of any non-trivial compression algorithm for a circuit class C subset of P/poly would imply the circuit lower bound NEXP not subset of C. This complements Williams's result [55] that any non-trivial circuit-sat algorithm for a circuit class C would imply a superpolynomial lower bound against C for a language in NEXP1.

关键词： average-case circuit lower bounds circuit-sat algorithms compression meta-algorithms natural property random restrictions shrinkage of de Morgan formulas

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